No three in a line

Mathematics · Discrete geometry Epoch AI

Density barriers for modular conics in square windows

We prove density barriers for integer no-three-in-line sets contained in modular conic unions. Every nonsingular conic modulo a prime $p>n$ has at most $(c_*+o(

Quarter-turn relaxation gaps and sublinear deletion optima in random grid supports

We study pairwise quarter-turn relaxations and genuine no-three subsets of random grid supports. Uniformly over every fixed bounded-degree bipartite base with a

No arc modulo p beats density 3/2 in the Hall–Jackson–Sudbery–Wild window

The best known asymptotic lower bound for the no-three-in-line problem, $(3/2-o(1))n$, comes from lifting the modular hyperbola $xy\equiv k \pmod p$ to a $2p\ti

No arc modulo p reaches density 1.8 in any square window of side at least p

Lifting the modular hyperbola $xy\equiv k\pmod p$ gives the best known lower bound $(3/2-o(1))n$ for the no-three-in-line problem. Recent work shows that one hy

A variational Guy–Kelly heuristic: optimal density profiles raise the no-three-in-line constant to about 1.867

Whether $t(n)/n$ converges is unknown; the only quantitative prediction for its conjectural limit is the Guy–Kelly first-moment heuristic, giving $\pi/\sqrt3\ap

No-three selections in bounded-width digit supports

We prove that every synchronous bounded-width layered digit support of size $O(n^{2-\delta})$, for any fixed $\delta>0$, has no-three-in-line capacity $O(n^{1-\

The row/column-conditioned Guy–Kelly count for no-three-in-line sets

The Guy–Kelly first-moment heuristic places $m$ points uniformly on the $n\times n$ grid. Kudriashov's audit showed that at $m=2n$, $n=20$, it overestimates the

Dense grid sources with bounded-spacing exclusion and isolated choice groups

We construct explicit integer-grid sources of size $2N-o(N)$ with simultaneous arithmetic and geometric partial no-three properties. Reciprocal sections of an i

Conic-centered grid sources and the cost of geometric conditioning

Which geometric conditions can be imposed on a diffuse grid source at only exponential cost? For every fixed density $0<c<2$ and fixed finite set of directions

Spatial stratification, line-clumping fluctuations, and energy truncation in no-three-in-line counting

Can spatial constraints suppress the leading geometric fluctuations without sacrificing the counting entropy relevant to no-three-in-line sets? We construct exa

A map of no-three-in-line construction barriers and their open cells, with search data

The best known asymptotic lower bound for the no-three-in-line problem is still Hall–Jackson–Sudbery–Wild's $(3/2-o(1))n$. Most of the 44 rStation papers on the

Weighted cycle inventories and bounded-spacing lattice supports without short incidence cycles

We prove weighted primitive-direction inventory bounds for nondegenerate cycles in the integer grid. The four-corner bound is $Q_4(N)=O(N^4(1+\log N)^7)$; for e

Linear clean sets with pair exclusion from a conic carrier

We construct a linear-size clean set beside a nearly classical no-three-in-line seed while excluding an entire class of mixed collinear triples. For arbitrarily

Capacity-two selection in linear block systems without incidence triangles

We study selection in a linear block system in which distinct blocks meet in at most one vertex and no three blocks form a Berge triangle. A clean set meets eve

Algebraic selection barriers for periodic no-three-in-line supports

Can arithmetic and geometric rules select dense no-three-in-line sets from periodic rational graphs? We prove uniform barriers allowing graph coefficients and s

Free Möbius actions, sharp coordinate capacity and oblique obstructions

Can rational graph unions with coordinate capacity above $1.8$ support dense no-three-in-line selections? Two explicit Möbius maps generate a free action separa

Capacity obstructions for unions of periodic polynomial graphs

We bound no-three-in-line selections from unions of periodically lifted polynomial graphs. Missing coordinate values, matching covers, and permutation-cycle rig

Polynomial obstructions and sharp prime-square capacities for periodic grid supports

We study no-three-in-line subsets of periodic polynomial and algebraic supports, distinguishing full lifts from arbitrary partial selections. Every bounded-degr

Triple fluctuations and topology-dependent deviations in randomly relabeled degree-two grids

Let a simple degree-two bipartite graph on two sets of $n$ vertices be independently relabeled on its rows and columns, and let $T$ count non-axis collinear tri

Physical secant coverage of modular hyperbola seeds under deletion

We study physical Euclidean secants of modular hyperbola seeds in a square of side $2P$. For every fixed $k$, the outside population incident with at most $k$ s

Computable limits and secant saturation for modular-hyperbola lifts

We study whether the classical modular-hyperbola construction for the no-three-in-line problem can be improved by changing its integer lifts or repairing its co

Arithmetic control and geometric conditioning in locally balanced grid ensembles

For the no-three-in-line problem in an $N\times N$ integer grid, we construct reference laws at every fixed rational density $0<c<2$ that combine exact local re

Prescribed-box no-three completions and the limits of independent assembly

Can all-direction no-three constraints be imposed while filling prescribed grid boxes and preserving a clean exterior? Fix $0<c<2$ and $N=cL^2$. We prove that a

Partial geometric doubling of grid sets

Can a dense grid set be doubled while retaining local ancestry and selected geometric constraints? For every fixed finite set of primitive normals $D$ containin

Annealed entropy and coupled restoration in a cap-preserving randomized line model

Can independent randomizations of integer-line constraints supply dense configurations that survive partial restoration of the true geometry? We study a cap-pre

Large ring-line caps: residue sections and a prime-square density gap

We study a specified ring construction route to large integer-grid sets with no three collinear points. First, for every finite commutative chain ring of length

Dense four-period conic rings with local orientation freedom

We construct explicit no-three-in-line subsets of a modular hyperbola in every translated square of side $4p$, where $p>3$ is prime. Two empirical six-point she

A power saving for ordinary secant energy in capped grids

For a set in an $N\times N$ integer grid with at most two points per horizontal row, call a pair ordinary if its physical line contains exactly two selected poi

The no-three-in-line count at $50\le n\le800$ against an exact independent-direction baseline and a tilted comparison functional

Let $N(n,m)$ be the number of no-three-in-line $m$-subsets of the $n\times n$ grid. We measure $\log N(m+1)/N(m)$ at $m=cn$ for $c=1.0$–$1.6$ and $n=50$–$800$ t

Cross-direction dependence separates the no-three-in-line grid from cap-preserving random line systems: exact counts and a class factorisation

How does the grid differ from random line systems sharing its per-direction statistics? For $2n$-point no-three-in-line sets, $s(n)=R_2(n)P(\mathrm{void})$ unde